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Phenotypic Variability of Polygenic Traits

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Phenotypic Variability of Polygenic Traits Heredity 59 (1987) 19—28 The Genetical Society of Great Britain Received 5 August 1986 Biochemical heterozygosity and phenotypic variability of polygenic traits Ranajit Chakraborty Center for Demographic and Population Genetics, University of Texas Graduate School of Biomedical Sciences, P.O. Box 20334, Houston, Texas 77225, U.S.A. Using the theory of additive genetic variability of a polygenic trait, it is shown than an individual's heterozygosity at the loci governing the trait cannot be determined accurately from observations on phenotypes alone. Furthermore, the negative association between heterozygosity and phenotypic variance, and a positive correlation of the frequency of the modal class of a phenotypic trait and the extent of heterozygosity can be explained by additive allelic effects. It is argued that while the number of heterozygous loci in an individual may not be a good indicator of the individual's genomic heterozygosity, there is evidence that some of the biochemical loci may reflect genetic variation at the loci controlling phenotypic polymorphism. Thus the observed relationship between biochemical heterozygosity and phenotypic variance may not constitute hard evidence of heterosis, overdominance, or associative overdominance. INTRODUCTION invoked to explain the observation of positive associations between the heterozygosity and the Allozymicvariations in natural populations, lack of phenotypic variability (see e.g., the referen- studied over the past two decades, have now con- ces cited in the above two reviews, and Livshits vincingly established that genetic variation is gen- and Kobyliansky, 1985). erally ubiquitous in almost all taxa. Recent The main features of the positive results of such molecular data showing genic variations at the association studies may be summarised into three DNA level have strengthened this view (Nei, 1987). observations. (a) When heterozygosity is measured Yet, the controversy regarding the mechanistic by the number of loci at which an individual is explanations for the production and maintainance heterozygous, increased heterozygosity is often of genetic polymorphism is still unresolved, since associated with a decreased phenotypic variability. direct searches for selection at any individual locus (b) The frequency of modal phenotypes is posi- have produced ambiguous results at best. More tively correlated with the heterozygosity levels of recently, therefore, attempts have been made to individuals. (c) Parental fitness is positively associ- relate the heterozygosity levels detected by multi- ated with the expected degree of heterozygosity ple loci with variabilities in morphological or among offspring. None of these findings are, fitness related traits in order to test heterotic effects however, universal, since exceptions are found in of the loci studied. Critical appraisals of the use repeated studies in the same population (Pierce of allozyme data in this regard have been made in and Mitton, 1982), and in studies involving popu- recent reviews of Mitton and Grant (1984) and lations of different evolutionary origin (e.g., Zouros and Foltz (1987). These reviews and their Gottlieb, 1977; Handford, 1980; Knowles and cited references indicate that the efforts to relate Mitton, 1980; Knowles and Grant, 1981; Mitton allozymic heterozygosity with phenotypic variabil- et al., 1981; Chakraborty et a!., 1986). ity do not always produce positive results. Yet, in In spite of these discordant results, the above many organisms, from plants to human, associ- features of association of enzymic heterozygosity ations between these two parameters suggest that with phenotypic variability demand satisfactory heterotic effects may be detectable by studying mechanistic explanation. The interpretation of interactions of multiple loci together. The concept selective differentials among individuals involve of developmental homeostasis has also been the assumption that the morphological traits used 20 R. CHAKRABORTY in determining phenotypic variability are geneti- A) and q, (for the allele B), where q =1—p1. The cally controlled, and the biochemical loci heterozygosity for each individual is measured by employed reflect genetic variation at the loci con- the number of heterozygous loci, if there are trolling such traits. It is true that genetic factors genotypic scoring techniques available for these are involved in morphological variation, and loci. Let Y denote the number of heterozygous fitness may be genetically controlled. Yet, the role loci in a random individual, and X denote the of non-genetic modifiers in determining genotypic value of the individual. We can write X phenotypes of such traits cannot be totally ignored. and Y as Furthermore, the task is more complicated, since the number and nature of genetic loci involved in x=xI+x2+. .+xn (1 a) morphological or fitness related traits are not and known, and their relationships with structural bio- chemical loci are not clear. Y=Y1+Y2+...+Yn, (ib) The purpose of this paper is to show that the where the bivariate distribution of X, and 1', is classification of individuals into different heterozy- given by gosity classes by phenotypes of a polygenic trait is quite error prone, even if the trait is under complete genetic control. It is also shown that the X1 Y, Genotype Probability increased frequency of modal phenotypes in highly heterozygous individuals (and the consequent 2 0 AA p decrease of phenotypic variability in them) is a 1 1 A,B1 2pq1 direct consequence of additivity of allelic effects 0 0 B.B1 q of a polygenic trait. Lastly, the question of inter- dependence of heterozygosity at biochemical and Under the assumption that the loci are indepen- phenotypic level is addressed by reviewing the dently segregating in the population, the joint dis- theory of predictability of genomic heterozygosity tribution of (X, Y) can be evaluated by the bivari- using electrophoretic markers, in view of the recent ate probability generating function method (Feller, comments of Zouros and Foltz (1987) and Smouse 1950); i.e., the probability that X =rand Y= k is (1986). given by Prob(X=r, Y=k) DISTRIBUTIONOF HETEROZYGOSITY BY = Coefficientof ss in G(s1, s2), PHENOTYPE CLASSES where Inrelation to the studies of morphological vari- ation and the heterozygous nature of individuals, G(s1, s2) =H(q+2pq1s1s2+ps), (2) it is common to identify the heterozygous status of individuals by their phenotypic scores for a for r =0,1,2, .. , 2n;k 0, 1,2,..., n, and s1, s2 heritable polygenic trait (see e.g., Livshits and are two arbitrary variables taking values between Kobyliansky, 1984; 1985; Kobyliansky and —1 and 1. Livshits, 1985). This is based on the supposition Note that the marginal distributions of X and that the phenotypic value of a morphological trait Y can be obtained by evaluating the coefficients is determined by polygenic effects of the loci of s in G(s1, 1) and s in G(1, s2), respectively. controlling the trait, and the allelic effects are Chakraborty (1981) provided a computational additive. Nevertheless, there has been no attempt algorithm for computing the distribution of Y, and to check this analytically. To address this question, the same technique also applies to X. Since for the let us consider a quantitative trait controlled by n quantitative phenotypes, the distribution of Y can- loci, at each of which there are two segregating not be generally observed directly, one might be alleles (say, A, and B, for the ith locus; i = interested in drawing inference regarding it by the 1,2,...,n). For simplicity, let us assume that the conditional distribution of Y (number of heterozy- allelic effects are all additive. Without loss of gen- gous loci) given an observation on X (the erality we can assume that the allelic affects of A phenotypic score). In practice, however, this is is one and the effect of B, is zero, for all i complicated by the effects of non-genetic environ- 1, 2,..., n. We shall further assume that the allele mental factors which may modify the genotypic frequencies at the ith locus are p (for the allele score X. For simplicity, if we assume that X is HETEROZYGOSITY AND PHENOTYPIC VARIANCE 21 directly observable and is unaffected by environ- Itis interesting to note that when all loci con- mental factors, the conditional distribution of Y trolling the quantitative trait X have the same allele given X =ris given by frequencies, the conditional distribution of Y given X does not depend on the allele frequency Prob(Y=kIX=r) p. Furthermore, since equation (6) is defined only Coefficientof ss in G(s1, s2) for even values of r —k,for a fixed value of r(0 — r 2n), k can take values from 0 to mm. (r, 2n —r), Coefficientof s in G(s, 1) such that r —kis always even. Equations (3) and for 0 r2n, and 0 k n. (6) are instructive enough to show that the For the case where the gene frequencies at all classification of individuals into heterozygosity loci are equal, i.e., p, =p for all I, it is easy to show classes by phenotypic scores can be quite mis- that leading. This is numerically demonstrated in table 1, where the conditional distribution of Y (the Prob (X =r)= (2n) prq2n_r (4a) number of heterozygous loci) for different genotypic values (X) are shown for the case where the allele frequencies are identical for all loci (the and exact value is irrelevant, since eqn. (6) is indepen- dent ofp). Prob(Y= k)=()(2pq)k(p2+q2). Two situations are illustrated with table 1, the (4b) number of loci (n) being 5 and 10. In both cases it is seen that the central phenotype (X = 5 and Furthermore,in this case the multi-locus 10) can have heterozygosity at less than half of the genotype of any individual can be represented by loci with probability approaching 1/4 or more. It atriplet(n1, n2, n3),where n1, n2,and n3represent can be shown from eqn. (3) or (6), that the mean the numbers of loci exhibiting genotypes AA, AB, number of heterozygous loci increases gradually and BB, respectively; with n1 + n2 + n3 = n.
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